A Partially Defined Game with Costs
Satoshi Masuya
TL;DR
This work extends cooperative game theory to partially defined games with costs, where obtaining the worth of unknown coalitions incurs a nonnegative cost that reduces the grand coalition value by $c_v(S)$. It develops a Shapley-like solution $\tilde{\phi}$ for PDGs with costs by combining extended PDG dividends on known coalitions with a residual cost term, and provides a formal axiom system that yields a unique allocation. It also introduces an exit rule for continuing examinations via an indicator function $\gamma$, including extreme variants and discussions of alternative stopping rules. Together, these results offer a principled framework for fair payoff allocation under information-gathering costs and relate PDGs with costs to existing restricted-cooperation concepts.
Abstract
The present study explores a problem that can be resolved by employing the notion of a partially defined cooperative game, yet cannot by using a restricted game. The following situation is considered: First, it is assumed that the worth of the grand and singleton coalitions are known. It takes some amount of costs to obtain worth of unknown coalitions. If it is performed, then the worth of the grand coalition is decreased by the value of a cost function. With the view point of fairness of a payoff allocation, we should examine coalitional worth as much as possible. However, we should stop examining coalitional worth at some point since total payoff is reduced by continuing the examinations. We name the new decision making problem a partially defined cooperative game with costs. The problem of a partially defined cooperative game with costs is finding the solution of partially defined cooperative games at each point and the best exiting rule of examinations of coalitional worth.